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G = C62.18C23  order 288 = 25·32

13rd non-split extension by C62 of C23 acting via C23/C2=C22

metabelian, supersoluble, monomial

Aliases: C62.18C23, Dic32⋊11C2, C4⋊Dic3⋊2S3, (C2×C12).15D6, Dic3⋊C4⋊15S3, C6.5(C4○D12), (C2×Dic3).6D6, Dic3⋊Dic3⋊30C2, C2.7(D12⋊S3), C6.33(D4⋊2S3), (C6×C12).176C22, C6.21(Q8⋊3S3), C32⋊1(C42⋊2C2), C6.D12.7C2, C6.11D12.4C2, C2.7(D6.6D6), (C6×Dic3).3C22, C2.10(D6.3D6), (C2×C4).16S32, (C3×C4⋊Dic3)⋊5C2, C22.77(C2×S32), C3⋊2(C4⋊C4⋊S3), (C3×Dic3⋊C4)⋊2C2, (C3×C6).58(C4○D4), (C2×C6).37(C22×S3), (C22×C3⋊S3).7C22, (C2×C3⋊Dic3).19C22, SmallGroup(288,496)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C62 — C62.18C23
Chief series
C1 — C3 — C32 — C3×C6 — C62 — C6×Dic3 — Dic32 — C62.18C23
Lower central
C32 — C62 — C62.18C23
Upper central
C1 — C22 — C2×C4

Generators and relations for C62.18C23
 G = < a,b,c,d,e | a6=b6=1, c2=e2=b3, d2=a3, ab=ba, ac=ca, dad-1=a-1, ae=ea, cbc-1=b-1, bd=db, be=eb, cd=dc, ece-1=a3b3c, ede-1=b3d >

Subgroups: 570 in 137 conjugacy classes, 44 normal (all characteristic)
C1, C2, C2, C3, C3, C4, C22, C22, S3, C6, C6, C2×C4, C2×C4, C23, C32, Dic3, C12, D6, C2×C6, C2×C6, C42, C22⋊C4, C4⋊C4, C3⋊S3, C3×C6, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C22×S3, C42⋊2C2, C3×Dic3, C3⋊Dic3, C3×C12, C2×C3⋊S3, C62, C4×Dic3, Dic3⋊C4, Dic3⋊C4, C4⋊Dic3, C4⋊Dic3, D6⋊C4, C3×C4⋊C4, C6×Dic3, C2×C3⋊Dic3, C6×C12, C22×C3⋊S3, C4⋊C4⋊S3, Dic32, C6.D12, Dic3⋊Dic3, C3×Dic3⋊C4, C3×C4⋊Dic3, C6.11D12, C62.18C23
Quotients: C1, C2, C22, S3, C23, D6, C4○D4, C22×S3, C42⋊2C2, S32, C4○D12, D4⋊2S3, Q8⋊3S3, C2×S32, C4⋊C4⋊S3, D12⋊S3, D6.6D6, D6.3D6, C62.18C23

Smallest permutation representation of C62.18C23
►On 48 points
Generators in S48
(1 2 3 4 5 6)(7 8 9 10 11 12)(13 14 15 16 17 18)(19 20 21 22 23 24)(25 26 27 28 29 30)(31 32 33 34 35 36)(37 38 39 40 41 42)(43 44 45 46 47 48)

(1 15 3 17 5 13)(2 16 4 18 6 14)(7 46 11 44 9 48)(8 47 12 45 10 43)(19 25 21 27 23 29)(20 26 22 28 24 30)(31 41 35 39 33 37)(32 42 36 40 34 38)

(1 44 17 7)(2 45 18 8)(3 46 13 9)(4 47 14 10)(5 48 15 11)(6 43 16 12)(19 40 27 32)(20 41 28 33)(21 42 29 34)(22 37 30 35)(23 38 25 36)(24 39 26 31)

(1 35 4 32)(2 34 5 31)(3 33 6 36)(7 30 10 27)(8 29 11 26)(9 28 12 25)(13 41 16 38)(14 40 17 37)(15 39 18 42)(19 44 22 47)(20 43 23 46)(21 48 24 45)

(1 30 17 22)(2 25 18 23)(3 26 13 24)(4 27 14 19)(5 28 15 20)(6 29 16 21)(7 32 44 40)(8 33 45 41)(9 34 46 42)(10 35 47 37)(11 36 48 38)(12 31 43 39)
 
G:=sub<Sym(48)| (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24)(25,26,27,28,29,30)(31,32,33,34,35,36)(37,38,39,40,41,42)(43,44,45,46,47,48), (1,15,3,17,5,13)(2,16,4,18,6,14)(7,46,11,44,9,48)(8,47,12,45,10,43)(19,25,21,27,23,29)(20,26,22,28,24,30)(31,41,35,39,33,37)(32,42,36,40,34,38), (1,44,17,7)(2,45,18,8)(3,46,13,9)(4,47,14,10)(5,48,15,11)(6,43,16,12)(19,40,27,32)(20,41,28,33)(21,42,29,34)(22,37,30,35)(23,38,25,36)(24,39,26,31), (1,35,4,32)(2,34,5,31)(3,33,6,36)(7,30,10,27)(8,29,11,26)(9,28,12,25)(13,41,16,38)(14,40,17,37)(15,39,18,42)(19,44,22,47)(20,43,23,46)(21,48,24,45), (1,30,17,22)(2,25,18,23)(3,26,13,24)(4,27,14,19)(5,28,15,20)(6,29,16,21)(7,32,44,40)(8,33,45,41)(9,34,46,42)(10,35,47,37)(11,36,48,38)(12,31,43,39)>;
 
G:=Group( (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24)(25,26,27,28,29,30)(31,32,33,34,35,36)(37,38,39,40,41,42)(43,44,45,46,47,48), (1,15,3,17,5,13)(2,16,4,18,6,14)(7,46,11,44,9,48)(8,47,12,45,10,43)(19,25,21,27,23,29)(20,26,22,28,24,30)(31,41,35,39,33,37)(32,42,36,40,34,38), (1,44,17,7)(2,45,18,8)(3,46,13,9)(4,47,14,10)(5,48,15,11)(6,43,16,12)(19,40,27,32)(20,41,28,33)(21,42,29,34)(22,37,30,35)(23,38,25,36)(24,39,26,31), (1,35,4,32)(2,34,5,31)(3,33,6,36)(7,30,10,27)(8,29,11,26)(9,28,12,25)(13,41,16,38)(14,40,17,37)(15,39,18,42)(19,44,22,47)(20,43,23,46)(21,48,24,45), (1,30,17,22)(2,25,18,23)(3,26,13,24)(4,27,14,19)(5,28,15,20)(6,29,16,21)(7,32,44,40)(8,33,45,41)(9,34,46,42)(10,35,47,37)(11,36,48,38)(12,31,43,39) );
 
G=PermutationGroup([[(1,2,3,4,5,6),(7,8,9,10,11,12),(13,14,15,16,17,18),(19,20,21,22,23,24),(25,26,27,28,29,30),(31,32,33,34,35,36),(37,38,39,40,41,42),(43,44,45,46,47,48)], [(1,15,3,17,5,13),(2,16,4,18,6,14),(7,46,11,44,9,48),(8,47,12,45,10,43),(19,25,21,27,23,29),(20,26,22,28,24,30),(31,41,35,39,33,37),(32,42,36,40,34,38)], [(1,44,17,7),(2,45,18,8),(3,46,13,9),(4,47,14,10),(5,48,15,11),(6,43,16,12),(19,40,27,32),(20,41,28,33),(21,42,29,34),(22,37,30,35),(23,38,25,36),(24,39,26,31)], [(1,35,4,32),(2,34,5,31),(3,33,6,36),(7,30,10,27),(8,29,11,26),(9,28,12,25),(13,41,16,38),(14,40,17,37),(15,39,18,42),(19,44,22,47),(20,43,23,46),(21,48,24,45)], [(1,30,17,22),(2,25,18,23),(3,26,13,24),(4,27,14,19),(5,28,15,20),(6,29,16,21),(7,32,44,40),(8,33,45,41),(9,34,46,42),(10,35,47,37),(11,36,48,38),(12,31,43,39)]])
 

42 conjugacy classes

class 1 2A2B2C2D3A3B3C4A4B4C4D4E4F4G4H4I6A···6F6G6H6I12A···12H12I···12P
order122223334444444446···666612···1212···12
size11113622446666121218182···24444···412···12

42 irreducible representations

dim11111112222224444444
type++++++++++++-+++
imageC1C2C2C2C2C2C2S3S3D6D6C4○D4C4○D12S32D4⋊2S3Q8⋊3S3C2×S32D12⋊S3D6.6D6D6.3D6
kernelC62.18C23Dic32C6.D12Dic3⋊Dic3C3×Dic3⋊C4C3×C4⋊Dic3C6.11D12Dic3⋊C4C4⋊Dic3C2×Dic3C2×C12C3×C6C6C2×C4C6C6C22C2C2C2
# reps11211111142681221222

Matrix representation of C62.18C23 ►in GL8(𝔽13)

120000000
012000000
001200000
000120000
00001000
00000100
00000001
0000001212
,
120000000
012000000
00100000
00010000
000012100
000012000
00000010
00000001
,
50000000
05000000
00050000
00800000
00000100
00001000
00000010
00000001
,
08000000
80000000
00500000
00050000
000012000
000001200
00000010
0000001212
,
01000000
120000000
00010000
00100000
000012000
000001200
00000010
00000001

G:=sub<GL(8,GF(13))| [12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,1,12],[12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[5,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,8,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,12,0,0,0,0,0,0,0,12],[0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1] >;
 

C62.18C23 in GAP, Magma, Sage, TeX 

C_6^2._{18}C_2^3
 
% in TeX
 
G:=Group("C6^2.18C2^3");
 
// GroupNames label
 
G:=SmallGroup(288,496);
 
// by ID
 
G=gap.SmallGroup(288,496);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,112,64,590,219,100,1356,9414]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^6=b^6=1,c^2=e^2=b^3,d^2=a^3,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1,a*e=e*a,c*b*c^-1=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e^-1=a^3*b^3*c,e*d*e^-1=b^3*d>;
 
// generators/relations
 

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